Time-averaged continuous quantum measurement

The theory of continuous quantum measurement allows to reconstruct the state $\rho_t$ of a system from a continuous stochastic measurement record $I_t$. However, this truly continuous-time signal $I_t$ is never available in practice. In experiments, one generally has access to its digitization, i.e., to a series of time averages $I_k$ over finite intervals of duration $\Delta t$. In this letter, we take this digitization seriously and define $\bar{\rho}_n$ as the best Bayesian estimate of the quantum state given (only) a digitized record $(I_1,\dots,I_n)$. We show that $\bar{\rho}_{n+1}$ can be computed recursively from $I_{n+1}$ and $\bar{\rho}_n$ using an exact formula. The latter can be evaluated numerically exactly, or used as the basis for a perturbative expansion into successive powers of $\sqrt{\Delta t}$. This allows reconstructing quantum trajectories in regimes of coarse $\Delta t$ where existing methods fail, estimating parameters at fixed $\Delta t$ without bias, and directly sampling digitized quantum trajectories with schemes of arbitrarily high order.